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33
Adaptive mesh refinement using wavepropagation algorithms for hyperbolic systems
 SIAM J. Numer. Anal
, 1998
"... Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a ..."
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Cited by 76 (8 self)
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Dedicated to Ami Harten for his many contributions and warm sense of humor. Abstract. An adaptive mesh refinement algorithm developed for the Euler equations of gas dynamics has been extended to employ highresolution wavepropagation algorithms in a more general framework. This allows its use on a variety of new problems, including hyperbolic equations not in conservation form, problems with source terms or capacity functions, and logically rectangular curvilinear grids. This framework requires a modified approach to maintaining consistency and conservation at grid interfaces, which is described in detail. The algorithm is implemented in the amrclaw package, which is freely available.
A wavepropagation method for conservation laws and balance laws with spatially varying flux functions
 SIAM J. Sci. Comput
, 2002
"... Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a ge ..."
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Cited by 66 (6 self)
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Abstract. We study a general approach to solving conservation laws of the form qt+f(q, x)x =0, where the flux function f(q, x) has explicit spatial variation. Finitevolume methods are used in which the flux is discretized spatially, giving a function fi(q) over the ith grid cell and leading to a generalized Riemann problem between neighboring grid cells. A highresolution wavepropagation algorithm is defined in which waves are based directly on a decomposition of flux differences fi(Qi)− fi−1(Qi−1) into eigenvectors of an approximate Jacobian matrix. This method is shown to be secondorder accurate for smooth problems and allows the application of wave limiters to obtain sharp results on discontinuities. Balance laws qt + f(q, x)x = ψ(q, x) are also considered, in which case the source term is used to modify the flux difference before performing the wave decomposition, and an additional term is derived that must also be included to obtain full accuracy. This method is particularly useful for quasisteady problems close to steady state. Key words. finitevolume methods, highresolution methods, conservation laws, source terms, discontinuous flux functions AMS subject classifications. 65M06, 35L65 PII. S106482750139738X
Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains
 In preparation; http://www.amath. washington.edu/~rjl/pubs/circles
, 2005
"... Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational do ..."
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Cited by 22 (6 self)
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Abstract. We describe a class of logically rectangular quadrilateral and hexahedral grids for solving PDEs in circular and spherical domains, including grid mappings for the circle, the surface of the sphere and the threedimensional ball. The grids are logically rectangular and the computational domain is a single Cartesian grid. Compared to alternative approaches based on a multiblock data structure or unstructured triangulations, this approach simplifies the implementation of numerical methods and the use of adaptive refinement. A more general domain with a smooth boundary can be gridded by composing one of the mappings from this paper with another smooth mapping from the circle or sphere to the desired domain. Although these grids are highly nonorthogonal, we show that the highresolution wavepropagation algorithm implemented in clawpack can be effectively used to approximate hyperbolic problems on these grids. Since the ratio between the largest and smallest grid is below 2 for most of our grid mappings, explicit finite volume methods such as the wave propagation algorithm do not suffer from the center or pole singularities that arise with polar or latitudelongitude grids. Numerical test calculations illustrate the potential use of these grids for a variety of applications including Euler equations, shallow water equations, and acoustics in a heterogeneous medium. Pattern formation from a reactiondiffusion equation on the sphere is also considered. All examples are implemented in the clawpack software package and full source code is available on the web, along with matlab routines for the various mappings.
The MoTICE: A new highresolution wavepropagation algorithm based on Fey’s Method of Transport
, 2000
"... Fey’s Method of Transport (MoT) is a multidimensional fluxvectorsplitting scheme for systems of conservation laws. Similarly to its onedimensional forerunner, the Steger–Warming scheme, and several other upwind finitedifference schemes, the MoT suffers from an inconsistency at sonic points when ..."
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Cited by 17 (3 self)
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Fey’s Method of Transport (MoT) is a multidimensional fluxvectorsplitting scheme for systems of conservation laws. Similarly to its onedimensional forerunner, the Steger–Warming scheme, and several other upwind finitedifference schemes, the MoT suffers from an inconsistency at sonic points when used with piecewiseconstant reconstructions. This inconsistency is due to a cellcentered evolution scheme, which we call MoTCCE, that is used to propagate the waves resulting from the fluxvectorsplitting step. Here we derive new firstorder and secondorderconsistent characteristic schemes based on interfacecentered evolution, which we call MoTICE. We prove consistency at all points, including the sonic points. Moreover, we simplify Fey’s wave decomposition by distinguishing clearly between a linearization and a decomposition step. Numerical experiments confirm the stability and accuracy of the new schemes. Owing to the simplicity of the two new ingredients of the MoTICE, its secondorder version is several times faster than that of the
A Wave Propagation Algorithm for Hyperbolic Systems on Curved Manifolds
"... An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, sh ..."
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Cited by 14 (0 self)
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An extension of the wave propagation algorithm first introduced by LeVeque [J. Comp. Phys. 131, 327353 (1997)] is developed for hyperbolic systems on a general curved manifold. This extension is important in a variety of applications, including the propagation of sound waves on a curved surface, shallow water flow on the surface of the Earth, shallow water magnetohydrodynamics in the solar tachocline, and relativistic hydrodynamics in the presence of compact objects such as neutron stars and black holes. As is the case for the Cartesian wave propagation algorithm, this new approach is second order accurate for smooth flows and highresolution shockcapturing. The algorithm is formulated such that scalar variables are numerically conserved and vector variables have a geometric source term that is naturally incorporated into a modified Riemann solver. Furthermore, all necessary onedimensional Riemann problems are solved in a locally valid orthonormal basis. This orthonormalization allows one to solve Cartesian Riemann problems that are devoid of geometric terms. The new method is tested via application to the linear wave equation on a curved manifold as well as the shallow water equations on part of a sphere. The proposed algorithm has been implemented in the software package clawpack and is freely available on the web.
Nonlinear Conservation Laws and Finite Volume Methods for Astrophysical Fluid Flow
 Computational Methods for Astrophysical Fluid Flow, 27th SaasFee Advanced Course Lecture Notes
, 1998
"... Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : ..."
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Cited by 14 (0 self)
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Contents 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1 Software : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.2 Notation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.3 Classification of differential equations : : : : : : : : : : : : : : : : : : : : : : : 7 2. Derivation of conservation laws : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 2.1 The Euler equations of gas dynamics : : : : : : : : : : : : : : : : : : : : : : : 13 2.2 Dissipative fluxes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Source terms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.4 Radiative trans
How to incorporate the springmass conditions in finitedifference schemes
 SIAM J. Scient. Comput
"... Abstract. The springmass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these springmass conditions in classical finited ..."
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Cited by 12 (11 self)
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Abstract. The springmass conditions are an efficient way to model imperfect contacts between elastic media. These conditions link together the limit values of the elastic stress and of the elastic displacement on both sides of interfaces. To insert these springmass conditions in classical finitedifference schemes, we use an interface method, the Explicit Simplified Interface Method (ESIM). This insertion is automatic for a wide class of schemes. The interfaces do not need to coincide with the uniform cartesian grid. The local truncation error analysis and numerical experiments show that the ESIM maintains, with interfaces, properties of the schemes in homogeneous medium.
Adaptive Mesh Refinement for conservative systems: multidimensional efficiency evaluation
"... Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro and magnetohydrodynamic simulations, AMR is used in combination with several shockcapturi ..."
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Cited by 12 (2 self)
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Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro and magnetohydrodynamic simulations, AMR is used in combination with several shockcapturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multidimensional calculations and only 1.5  8 % overhead is observed. For AMR calculations of multidimensional magnetohydrodynamic problems, several strategies for controlling the r B = 0 constraint are examined. Three source term approaches suitable for cellcentered B representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing LaxFriedrichs method on the coarser levels. This leveldependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.
An Unconditionally Stable Method For The Euler Equations
 J. COMPUT. PHYS
, 1999
"... We discuss how to combine a front tracking method with dimensional splitting to solve numerically systems of conservation laws in two space dimensions. In addition we present an adaptive grid refinement strategy. The method is unconditionally stable and allows for moderately high cfl numbers (typ ..."
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Cited by 10 (4 self)
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We discuss how to combine a front tracking method with dimensional splitting to solve numerically systems of conservation laws in two space dimensions. In addition we present an adaptive grid refinement strategy. The method is unconditionally stable and allows for moderately high cfl numbers (typically 14), and thus it is highly efficient. The method is